Edge Split disconnecting the edge
Yes.  That's what an edge split modifier does.  If you don't want to disconnect the edges, don't use an edge split modifier.  If you're trying to use an edge split modifier to create hard edges, there are other techniques (mark sharp, autosmooth, which it appears you're already using); if you need an edge split modifier for an export somewhere, don't apply it until after you weight, if it's screwing you up.